Inelastic neutron scattering is used to investigate the collective magnetic excitations of the high-temperature superconductor-parent antiferromagnet La2CuO4. We find that while the lower energy excitations are well described by spin-wave theory, including one- and two-magnon scattering processes, the high-energy spin waves are strongly damped near the (1/2, 0) position in reciprocal space and merge into a momentum dependent continuum. This anomalous damping indicates the decay of spin waves into other excitations, possibly unbound spinon pairs.